Heavy and leafy trees

The MAXIMUM WEIGHTED LEAF TREE problem consists of, given a connected graph G and a weight function w: V (G) → Q+, finding a tree in G whose weight on the leaves is maximized. The variant that requires the tree to be spanning is at least as hard to approximate as the maximum independent set problem. If all weights are unitary, it turns into the well-known problem of finding a spanning tree with maximum number of leaves, which is NP-hard, but is in APX. No further innapproximability result is known for MAXIMUM WEIGHTED LEAF TREE, and the best approximation is an O(lg n)-approximation. In this paper, we present an O(lg W)-approximation where W is the maximum weight divided by the minimum weight that appears on the vertices.